#### 3.6 Network Equations

For a network with buses, all constant impedance elements of the model are incorporated into a complex bus admittance matrix that relates the complex nodal current injections to the complex node voltages : (3.8)

Similarly, for a network with branches, the system branch admittance matrices and relate the bus voltages to the vectors and of branch currents at the from and to ends of all branches, respectively: If is used to denote an operator that takes an vector and creates the corresponding diagonal matrix with the vector elements on the diagonal, these system admittance matrices can be formed as follows: The current injections of (3.8)–(3.10) can be used to compute the corresponding complex power injections as functions of the complex bus voltages : The nodal bus injections are then matched to the injections from loads and generators to form the AC nodal power balance equations, expressed as a function of the complex bus voltages and generator injections in complex matrix form as (3.17)